Carrier and nerve theorems in the extension theory

Abstract

Abstract. We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equivalent to the underlying space.Next we prove a nerve theorem for a class of spaces with uniformly bounded extension dimension. In particular we prove that the canonical map from a separable metric n-dimensional space into the nerve of its weakly regular open cover induces isomorphisms on homotopy groups of dimensions less than n.

A Unified View on the Functorial Nerve Theorem and its Variations

“…The original work of Borsuk [Bor48] proves a nerve theorem for finite closed covers of metric spaces such that all non-empty intersections of cover elements are absolute retracts (hence contractible). Much more recently, Nagórko [Nag07] proves a nerve theorem for locally finite, locally finite dimensional, star-countable closed covers of normal spaces such that all non-empty intersections of cover elements are absolute extensors for metric spaces. Given a good cover of a finite simplicial complex by subcomplexes, Barmak [Bar11] proves that the simplicial complex and the nerve have the same simple homotopy type.…”

Local $k$-connectedness of an inverse limit of polyhedra

“…Carrier Theorem ( [13]). Assume that C : F → G is a carrier such that F is a cover of a space X and G is an AE(X)-cover of another space.…”

“…A Nerve Theorem in its abstract form states that if two spaces admit isomorphic AE(n)-covers, then they are weak n-homotopy equivalent. For general spaces, some local finiteness and dimension restrictions are placed on the covers [13]. In our case we need such a theorem without these restrictions, which we are able to prove for polyhedra covered by subcomplexes.…”

“…The main tool used in the proof is Theorem 3.5, which when applied to a canonical map into the nerve of a cover [13] implies the usual Nerve Theorem.…”

Vietoris thickenings and complexes have isomorphic homotopy groups

“…In Section 4 we modify this good cover that is not open in order to obtain an open cover that is only "good up to level n." This open cover is what we need in Section 5 to prove Theorem 1 that V m (U) and V(U) have isomorphic homotopy groups. Indeed, the main tool we use in our proof is different versions of the nerve theorem, including versions by Nagorko [35] and the third author [45]. The proof of Theorem 1 relies on the fact that V m (U) is locally contractible, which we prove in Theorem 2 in Section 6.…”